How do you sketch a parabola without computing a whole table of values?

The leading coefficient sets the direction

A quadratic is written y equals ax squared plus bx plus c, with a not zero. The sign of a is the first thing to check.

If a is positive the parabola opens upwards and the vertex is a minimum. If a is negative it opens downwards and the vertex is a maximum. A maximum question on the paper always arrives with a negative a, and the reverse.

c is the value when x is zero, which is the y-intercept. It is read straight off the equation with no calculation at all.

The vertex and the axis of symmetry

The x value of the vertex is minus b over 2a. Substitute it back into the function to get the y value, and the vertex is complete.

The vertical line through the vertex is the axis of symmetry. The parabola is mirrored in it: any two points at equal distances on either side sit at exactly the same height.

That gives a useful shortcut. If the two zeros are known, the x value of the vertex is their average. It is faster than the formula, and in an exam that is time.

Zeros and the extreme-value question

The points where the curve meets the x-axis are the solutions when y equals zero. Solve with the quadratic formula, or by factorising when that is available.

There may be two such points, one where the parabola touches the axis, or none at all when the whole curve sits above or below it. The discriminant — b squared minus 4ac — tells you which case you are in before you start.

A maximum or minimum question in a word problem is answered entirely at the vertex. Its x value answers when or how many; its y value answers what the greatest profit or height is. Notice which of the two was asked for.

Worked examples

1. Find the vertex of y equals x squared minus 6x plus 5.

   1. a is 1 and b is minus 6
   2. x of the vertex: minus b over 2a, which is 6 over 2
   3. x equals 3, substitute back
   4. 9 minus 18 plus 5

   Answer: The vertex is (3, minus 4), and it is a minimum because a is positive

2. Find where that parabola meets the axes.

   1. With the y-axis: set x to zero, giving 5
   2. With the x-axis: solve x squared minus 6x plus 5 equals zero
   3. Factorise: (x minus 1) times (x minus 5)

   Answer: It meets the y-axis at (0, 5) and the x-axis at (1, 0) and (5, 0)

3. A shop's profit is y equals minus 2x squared plus 40x minus 50. What is the maximum profit?

   1. a is negative, so the vertex is a maximum
   2. x of the vertex: minus 40 over minus 4, which is 10
   3. Substitute: minus 200 plus 400 minus 50

   Answer: The maximum profit is 150, reached when x is 10

Common mistakes

Dropping the minus sign in the vertex formula

The formula is minus b over 2a. When b is already negative the double negative flips the sign, and skipping it puts the vertex on the wrong side of the axis.

Answering with the x value when asked for the maximum

The vertex's x value says when, and its y value says how much. A question about the greatest profit wants y; a question about the number of items wants x.

Assuming every parabola has two zeros

If the discriminant is negative there is no x-intercept at all, and the curve lies entirely above or entirely below the axis. Finding yourself taking the square root of a negative number is the signal.

More in Mathematics, three units

• How do you turn a word problem into an equation without going wrong at the start?
• When do you use a distance-speed-time table and when the part done in one hour?
• How do you find the equation of the line through two given points?
• Why does a 10 percent rise followed by a 10 percent fall not return the original price?
• Two classes share an average — what else do you need to compare them?
• How do you find an average from a frequency table rather than from a list?